Beam Deflections

Moment Sign Convention

is measured the deflection of the beam

Θ is the slope for small angles

Differential equations of the deflection curve • Beams with small angles of rotation, and small deflection

• The structures encountered in everyday life, such as buildings, automobiles, aircraft, ships undergo relatively small changes in shape while in service. Therefore, we assume small angles of rotation and very small deflections

Differential equation of the deflection curve, v

M is bending moment

EI is commonly called flexural rigidity

2

21

dx

vd

EI

M

dx

vd

2

2

Differential equation of the curve with radius,

EI

M

dx

vd

2

2

dxEI

M

dx

dv

dxdxEI

Mv

Slope ~ (small )

Deflection, v

Differential equations of the deflection curve

Differential equations of the deflection curve

• Thus for prismatic beams

BENDING – MOMENT EQUATION

SHEAR – FORCE EQUATION LOAD EQUATION

Moment and deflection curve

Two Main Choices in Deflection Determination

• Find moment as function of beam length, x

– Integrate Moment twice

• Use superposition of previously solved problems

– Tables of common loadings, for easy look up

– Use linear superposition for combined loads, adding or subtracting as approptiate

Example: Find deflection in middle

Example solution: 1 + 2

1

2